TANGENT TO THE RICCI FLOW
MATTHIAS ERBAR AND NICOLAS JUILLET
Abstract. We study a transformation of metric measure spaces in- troduced by Gigli and Mantegazza consisting in replacing the original distance with the length distance induced by the transport distance between heat kernel measures. We study the smoothing effect of this procedure in two important examples. Firstly, we show that in the case of some Euclidean cones, a singularity persists at the apex. Secondly, we generalize the construction to a sub-Riemannian manifold, namely the Heisenberg group, and show that it regularizes the space instantaneously to a smooth Riemannian manifold.
1. Introduction
There are many ways to deform a Riemannian manifold into a singular met- ric space as discussed for instance in the influential essay of Gromov [18]. We are interested in the opposite question whether there exists a deformation, intrinsically defined for a wide class of metric spaces that instantaneously turns the space into a Riemannian manifold. In this paper, we investigate a method that has been introduced by Gigli and Mantegazza [17]. We ex- amine its regularization properties in two important cases: Euclidean cones and the Heisenberg group. These are emblematic examples of Alexandrov spaces and subRiemannian spaces respectively. We also discuss normed vector spaces where the transformation turns out to be the identity as an example of Finsler structures.
Before we state our results we briefly explain the main features of the con- struction of Gigli and Mantegazza which is based on the interplay of optimal transport and Ricci curvature. The starting point is a metric measure space (X, d, m) on which a reasonable notion of heat kernel can be defined. For t > 0 a new distance dt(x, y) is defined as the length distance induced by the L2 Wasserstein distance built from dbetween the heat kernel measures centered at xand y.
The striking feature of this approach is the following main result of [17]:
When (X, d, m) is a Riemannian manifold then dt is induced by a smooth metric tensor gt that is tangent to the Ricci flow, i.e. ∂t|t=0gt = −2 Ric in a weak sense. Gigli and Mantegazza then generalize this construction to metric measures spaces with generalized Ricci curvature lower bounds, namely the RCD condition, which ensures existence of a well-behaved heat kernel. This can be seen as a first step into constructing a Ricci flow for
2010 Mathematics Subject Classification. Primary 53C44; Secondary: 49Q20, 51F99, 51K10, 53C17.
Key words and phrases. Ricci flow, optimal transport, Euclidean cone, Heisenberg group.
1
non-smooth initial data. A related synthetic characterization of super-Ricci flows based on optimal transport has been obtained by McCann and Topping [24].
One can think of dt as a sort of convolution of the original distance with the heat kernel. Having the smoothing effect of the heat equation and Ricci flow in mind, one might expect that this procedure gives a canonical way of regularizing the metric measure space.
A first study of the regularizing effects of the Gigli-Mantegazza flow has been performed by Bandara, Lakzian and Munn [6] in the case where the distance d is induced by a metric tensor with low regularity and isolated conic singularities. It is shown that dt is induced by a metric tensor with at least the same regularity away from the original singular set. The question, what happens at the singularities has been left unanswered.
In the present paper, we give an answer showing that conic singularities can persist under the Gigli-Mantegazza transformation. We analyse in detail the transformation for two specific Euclidean cones of angleπandπ/2. Our results are the following (see Theorem 3.11 and Proposition 3.10 below).
Theorem 1.1. Let C(π) be the two-dimensional Euclidean cone of angle π and d its distance. For every t > 0 the convoluted distance dt has a conic singularity of angle √
2π at the apex.
As t goes to zero, the metric space (C(π), dt) tends to (C(π), d) pointwise and in the pointed Gromov–Hausdorff topology. As t goes to infinity, it tends to the Euclidean cone of angle √
2π in the pointed Gromov–Hausdorff topology.
In fact, it turns out that for fixed θ >0 all spaces (C(θ), dt) for t > 0 are isometric up to a multiplicative constant. An isometry is induced by the radial dilation x ∈ C(θ) 7→ t−1/2x. Our second result shows that for the cone of angleπ/2 the behavior of the singularity is even worse (see Theorem 3.17 and Proposition 3.16 below).
Theorem 1.2. Let C(π/2) be the two dimensional Euclidean cone of an- gle π/2 and d its distance. For every t > 0, the distance dt has a conic singularity of angle zero at the apex.
As t goes to zero, the metric space(C(π/2), dt) tends to (C(π/2), d) point- wise and in the pointed Gromov–Hausdorff topology. As tgoes to infinity, it tends to R+ with the Euclidean distance in the pointed Gromov–Hausdorff sense.
The reason why we focus on these two specific cones is that they can be conveniently represented as quotients of R2 under rotation by π and π/2 respectively. It turns out that the convoluted distance dt is the length dis- tance induced by the L2 Wasserstein distance between a mixtures of two (respectively four) rotated copies of Gaussian measures with variance 2t.
A corollary of the previous theorem is that the space (C(π/2), dt) is not an Alexandrov space even though C(π/2) is. In fact, in Alexandrov spaces a triangle with one angle zero is flat, which is wrong for (C(π/2), dt). This negative result has to be compared to positive results by Takatsu [31], where it is shown that the subspace made of all Gaussian measures in the Wasser- stein space over Euclidean space is an Alexandrov space. Note moreover,
that the Wasserstein space over a non-negatively curved Alexandrov space is again a non-negatively curved Alexandrov space [30, Proposition I.2.10] and that many subspaces of finite dimensional Alexandrov spaces are known to be Alexandrov spaces, for instance convex hypersurfaces in Euclidean spaces or Riemannian manifolds of sectional curvature bounded below [1,8,25].
Given the relation of the Gigli–Mantagazza flow with the Ricci flow, the convergence of dt to the original cone distance d has to be compared with the fact that any Euclidean cones of dimension 2 can be obtained as the backward limit of classical solutions to the Ricci flow [13, Chapter 4.5]. See also [29,14] for related results in higher dimension.
Our second contribution in this paper is an investigation of the Gigli–
Mantegazza flow applied to the first Heisenberg group equipped with the Carnot-Carath´eodory distance. The Heisenberg group is one of the sim- plest examples of a non trivial Carnot group, i.e a nilpotent stratified Lie groups with a left-invariant metric on the first strata, and of a non triv- ial subRiemannian manifold. These classes are of course connected: As proved by Bella¨ıche [7], the tangent cones at points of subRiemannian spaces are Carnot groups. The differentiable structure of the Heisenberg group is the one of R3 and the group structure is given in coordinates (x, y, u) by (x, y, u).(x0, y0, u0) = (x+x0, y+y0, u+u0+ (1/2)(xy0−x0y)).
The Carnot–Carath´eodory distance is obtained by minimizing the length of curves that are tangent to the 2-dimensional horizontal subbundle spanned by X=∂x−y2∂u and Y =∂y +x2∂u. A standard way to approximate this distance is to consider for ε >0 the Riemannian distancedRiem(ε) obtained by considering X, Y, ε∂u as an orthonormal frame. In fact, this penalization principle permits to see any subRiemannian manifold as a limit of Riemann- ian manifolds. Note that (H, dcc) does not satisfy a generalized lower Ricci curvature bound in the sense of the RCD condition. Therefore we slightly generalize the construction in [17] and obtain the following result (see The- orem 4.6 and Proposition 4.9 below).
Theorem 1.3. Let (H, dcc) be the first Heisenberg group equipped with the Carnot–Carath´eodory distance. For t > 0, the convoluted distance dt co- incides with KdRiem(κ√t), for some constants K, κ satisfying K ≥ 2 and K/κ <√
2.
As tgoes to zero the distancedt converges toKdcc pointwise. In the pointed Gromov–Hausdorff topology the space (H, dt) converges to(H, dcc).
The striking part of the theorem is that also non-horizontal curve can have finite length after lifting them to the Wasserstein space built from dcc via the heat kernel and thus dt becomes a Riemannian distance. We believe that this behavior also holds for more general contact manifolds. However, let us stress the fact that even for the Heisenberg group the distancedtdoes not converge pointwise to dcc as t goes to zero. Convergence in pointed Gromov–Hausdorff sense only holds due to the high amount of symmetry of the space, in particular, due to the fact that the dilation (x, y, u) 7→
(Kx, Ky, K2u) is an isometry between (H, Kdcc) and (H, dcc). The Gromov–
Hausdorff convergence probably does not hold for generic contact manifolds of dimension 3 with a subRiemannian metric on the nonholonomic contact
distribution. Finally, note that also the Heisenberg group can be obtained as a backward limit of classical solution to the Ricci flow as was shown by Cao and Saloff-Coste [11].
Three sections follow this introduction. The next section contains the con- struction of the convoluted distance dt in a general setting. As a first ex- ample we discuss the case of normed spaces. In Section 3 we establish our results on the Euclidean cones C(π) and C(π/2). Section 4 is devoted to the Heisenberg group.
Acknowledgements. The authors would like to thank Michel Bonnefont, Thomas Richard and Andr´e Schlichting for stimulating discussions on this work and related topics. Part of this work was accomplished while the authors were enjoying the hospitality of the Hausdorff Research Institute for Mathematics in Bonn during the Junior Trimester Program on Optimal Transport. They would like to thank HIM for its support and the inspiring atmosphere. M.E. gratefully acknowledges support by the German Research Foundation through the Collaborative Research Center 1060The Mathemat- ics of Emergent Effects and the Hausdorff Center for Mathematics. N.J. is partially supported by the Programme ANR JCJC GMT (ANR 2011 JS01 011 01).
2. Construction of the flow
In this section we present the construction of the convoluted distance dt in a general framework. The reason is that the framework of RCD spaces considered in [17] (see subsection 2.3) does not cover the Heisenberg group.
Moreover, unlike in [17] the spaces of the present paper are non-compact 2.1. Preliminaries. Let (X, d) be a Polish metric space. Recall that for p≥1 a curve (γ)t∈[0,T] in (X, d) is called p-absolutely continuous, for short γ ∈ACp [0, T],(X, d)
, if there exist a functionm∈Lp(0, T) such that for any 0≤s≤t≤T:
d(γ(s), γ(t)) ≤ Z t
s
m(r) dr .
For p = 1, we may simply call it an absolutely continuous curve. In this case the metric derivative defined by
|γ˙s| = lim
h→0
d(γs+h, γs) h
exists for a.e. s∈(0, T) and is the minimalm as above, see [2, Thm. 1.2.1].
Lipschitz curves with respect to a distance d are called d-Lipschitz curves, they are locally p-absolutely continuous for everyp≥1.
We denote by P(X) the set of Borel probability measures. The subset of measures with finite second moment, i.e. satisfying
Z
d(x0, x)2dµ(x)<∞
for some, hence anyx0∈Xwill be denoted byP2(X). Givenµ, ν ∈P2(X) their L2-Wasserstein distance is defined by
W(µ, ν) = inf
π
s Z
d(x, y)2 dπ(x, y),
where the infimum is taken over all couplings π of µ and ν. Recall that P2(X), W
is again a Polish metric space. Sometimes we will write WX
or W(X,d) to avoid confusion about the underlying metric space (X, d).
2.2. Construction of the flow. Recall that (X, d) is a metric Polish space.
Let us assume in addition that it is proper, i.e. closed balls are compact, and that it is a length space, i.e. we have
d(x, y) = inf
γ
Z T
0
|γ˙s|ds ,
where the infimum is taken over all absolutely continuous curvesγ connect- ingx toy. Notice that (X, d) is in fact geodesic, i.e. each pair of points can be joint by a curve whose length equals d(x, y).
The construction is based on a family of maps from X toP2(X) satisfying some properties that we list now. One should keep in mind that in the examples coming later the points are mapped to heat kernel measures.
Assumption 2.1. There exists a family (ιt)t≥0 of maps ιt :X → P2(X) with the following properties:
• ι0(x) =δx for allx∈X,
• ιt is injective for all t≥0,
• ιtis Lipschitz, more precisely, there exist constants Ct>0such that W ιt(x), ιt(y)
≤ Ctd(x, y) ∀x, y∈X , (2.1) and t7→Ct is locally bounded from above,
• the curve [0,∞)3t7→ιt(x) is continuous with respect to W for all x∈X.
We introduce a new family of distance functions det : X×X → [0,∞) for t≥0 given by
det(x, y) = W ιt(x), ιt(y) .
As W is a distance it follows from the injectivity of ιt that det is also a distance. It is the chord distance induced by the embedding ιt. The main object of study here will be the corresponding arc distance, i.e. the length distance induced by det, denoted bydt. More precisely, we define for t ≥0 and x, y∈X:
dt(x, y) = inf
γ
Z T
0
|γ˙s|tds , (2.2) where the infimum is taken over all curvesγ ∈AC [0, T]; (X,det)
such that γ0 = x, γT = y and |γ˙s|t denotes the metric derivative with respect to det. Note that (2.1) implies that
dt(x, y) ≤ Ctd(x, y) ∀x, y∈X . (2.3)
Indeed, for any curve (γs)sthat is absolutely continuous with respect todits metric derivative with respect to dis bounded above as|γ˙s|t≤Ct|γ˙s|. The claim then follows by integrating in sand taking the infimum over all such curves (γs)s noting that they are also absolutely continuous with respect todet and that (X, d) is a length space.
Remark 2.2. This construction is slightly different from the one in [17], where the infimum in the definition ofdtis taken overγin AC [0, T]; (X, d) which is a subset of AC [0, T]; (X,det)
by the Lipschitz assumption (2.1).
Allowing curves in the latter larger class will be crucial when applying the construction in the case of the Heisenberg group in Section 4. In the case of the Euclidean conesC(π), C(π/2) discussed in Section 3, we show in Lemma 3.6 that the infima over both classes of curves agree so that we are consistent with the construction in [17].
Remark 2.3. Note that the value of the infimum in (2.2) does not change, if we restrict the infimum to det-Lipschitz curves. Indeed, the right hand side of (2.2) is invariant by reparametrizreparametrizationation and every absolutely continuous curve can be reparametrized as a Lipschitz curve, see for instance [2, Lem. 1.1.4].
We can reformulate the definition of dt as follows. Given an absolutely continuous curve (γs)s∈[0,T] in (X,det) we obtain an absolutely continuous curve (µtγs)s∈[0,T] in P2(X), W
by setting µtγs =ιt(γs). Then we have dt(x, y) = inf
γ
Z T
0
|µ˙tγs|ds ,
where|µ˙tγs|denotes the metric derivative with respect toW. Another equiv- alent formulation is
dt(x, y) = inf sup
N−1
X
i=0
det(γsi, γsi+1) = inf sup
N−1
X
i=0
W(µtγsi, µtγsi+1), (2.4) the supremum being taken over all partitions 0 = s0 < s1 <· · · < sN = 1 and the infimum over all continuous curves (γs)s∈[0,1] connecting x toy.
In this general setup we have the following continuity properties.
Proposition 2.4. For all x, y∈X, the curve [0,∞)3t7→ det(x, y) is con- tinuous and the curve t7→ (X,det) is continuous with respect to the pointed Gromov–Hausdorff convergence. Moreover, assume in addition to Assump- tion 2.1 that bounded sets in (X,det) are bounded in (X, d). Then the dis- tances det and dt induce the same topology as the original distance d.
Proof. We first prove the convergence statement. Let (tn)n converge to t.
As an immediate consequence of Assumption 2.1 we have that detn(x, y) → det(x, y) for fixed x, y ∈ X. Moreover, by (2.1), for each compact set K in (X, d) the functions detn(·,·) are equicontinuous on K×K. Thus, they converge uniformly todet(·,·). This readily yields the convergence of (X,detn) to (X,det) in the pointed Gromov–Hausdorff sense. Now, we turn to the second statement. First, recall from (2.3) that det ≤ dt ≤ Ctd. Thus, it
suffices to show that for any sequence (xn)n, and element x of X with det(x, xn) → 0 as n → ∞ we also have that d(xn, x) → 0. By assumption, the sequence xn is bounded in (X, d). Thus, up to taking a subsequence we can assume that d(xn, x0)→ 0 for somex0 ∈X. Hence, also det(xn, x0)→0 and we infer thatx0 =x. This being independent of the subsequence chosen, we conclude that the full sequence xn converges tox in (X, d).
Remark 2.5. We proved the continuity of the mapt7→det(x, y). The conti- nuity of t7→ dt(x, y) fails for the Heisenberg group at t= 0 as we will see in Section 4. This is in contrast to [17, Thm. 5.18] where right-continuity of this map is shown. Note however, that the Heisenberg group does not satisfy the RCD condition and our construction is slightly different in this case, see Remark 2.2.
2.3. Riemannian manifolds and RCD spaces. In [17] the preceding construction has been introduced and studied in the case where (X, d) is a Riemannian manifold or more generally a metric measure spaces satisfying the Riemannian curvature-dimension condition for some curvature param- eter K ∈ R, denoted by RCD(K,∞). For short we call such spaces RCD spaces. In both cases the embeddingιt is constructed using the heat kernel.
Let us briefly recall the main results in [17].
Let (X, g) be a smooth compact and connected Riemannian manifold with metric tensorgand letdand vol be the associated Riemannian distance and volume measure. One can define a map ιt:X →P2(X) be settingιt(x) = νxt, where νxt(dy) =pt(x, y) vol(dy) is the heat kernel measure, i.e. pt(·,·) is the fundamental solution to the heat equation onX. It can be verified that Assumption 2.1 and Proposition 2.4 hold in this case.
Gigli and Mantegazza prove that the distances dt are induced by a family of smooth metric tensors (gt)t≥0 and that this flow of tensors is initially tangent to the Ricci flow [17, Prop. 3.5,Thm. 4.6]. More precisely, for every geodesic (γs)s∈[0,1] with respect to g=g0:
d
dtgt( ˙γs,γ˙s)
t=0 = Ric( ˙γs,γ˙s) for almost everys∈(0,1),
where Ric denotes the Ricci tensor of g. Gigli and Mantegazza then gen- eralize the construction for the initial data being a metric measure space satisfying the RCD(K,∞). Since we do not work in this general setting, we will describe it only briefly. For more details on RCD spaces we refer to [3,4].
Roughly speaking, RCD spaces form a natural class of metric measure spaces that can be equipped with a canonical notion of Laplace operator and a well behaved associated heat kernel. The RCD(K,∞) is a reinforcement of the curvature-dimension condition CD(K,∞) introduced by Lott–Villani and Sturm [23, 30] as a synthetic definition of a lower bound K on the Ricci curvature for a metric measure space (X, d, m). The condition CD(K,∞) asks for the relative entropy
Ent(µ) = Z
ρlogρdm , forµ=ρm∈P2(X)
to be K-convex along Wasserstein geodesics, i.e.
Ent(µs)≤(1−s) Ent(µ0) +sEnt(µ1)−K
2s(1−s)W(µ0, µ1)2 . The RCD(K,∞) condition requires in addition that the ‘heat flow’ obtained as the Wasserstein gradient flow of the entropy in the spirit of Otto [28] is linear. This excludes e.g. Finslerian geometries. It is a deep insight that the two requirements can be encoded simultaneously in the following property (which we take as a definition of RCD spaces for the purpose of this paper).
Theorem 2.6 (Definition of the RCD spaces through the EVI [4, Thm.
5.1]). Let K be a real number. The metric measure space (X, d, m) satisfies the Riemannian curvature-dimension condition RCD(K,∞) if and only if for every µ ∈P2(X) there exist an absolutely continuous curve (µt)t≥0 in (P2(X), W) starting fromµ in the sense that W2(µ, µt)→0 as t→0 and solving the Evolution Variational Inequality(in short EVI) of parameterK, i.e. for all ν ∈P2(X) such thatEnt(χ|m)<∞ and a.e. t >0:
d dt
1
2W(µt, χ)2+K
2W(µt, χ)2 ≤Ent(χ)−Ent(µt).
In fact, the solution µt to the EVI is unique and, putting Htµ = µt, one obtains a linear semigroup on P2(X) which is called the heat flow (acting on measures) in X. The construction in [17] then proceeds as presented in Section 2 by choosing the map ιt : X → P2(X) to be ιt(x) = Htδx. A natural example of RCD spaces are Euclidean cones, see [21].
2.4. Normed spaces. For an example that can be studied rapidly and is rather different let us consider the flow for Rn equipped with a norm k·k. Indeed, the metric measure space (Rn,k·k,Leb) satisfies the condition CD(0,∞) but does not satisfy RCD(0,∞) unlessk·kis induced by an inner product. It is possible to consider in this setting a non-linear heat equation, driven by a non-linear Laplace operator, see [26] for the a study in the much more general setting of Finsler manifolds. However, for a non-Hilbert norm there is no canonical choice of a heat kernel, i.e. a solution starting from a Dirac mass since contraction of the heat flow fails [27]. Note however, that a particular solution is given by the appealing formula [27, Example 4.3]
ft(x) = C 4πtexp
−kxk2 4t
,
whereC is a normalization constant. Hence a choice satisfying Assumption 2.1 isιt(x) =ft(·−x) Leb. Any other reasonable choice should be translation invariant. Let us show that in this case the distance dt coincides with the original one, i.e.dt(x, y) =kx−yk. Indeed, considerιt:x7→(τx)#νt where νt ∈ P2(Rd,k·k) is a measure and τx the translation by x. It is easily checked using Jensen’s inequality on the convex function (u, v)7→ ku−vk2 that W(Rn,k·k) ιt(x), ιt(y)
= kx−yk. The translation τy−x is an optimal map, in other words (τx, τy)#νt is an optimal coupling. Since the original distance was already a length distance we finddt(x, y) =det(x, y) =kx−yk.
Hence the flow leaves the space invariant and does not regularize it to a Riemannian manifold.
Remark 2.7. We stress that the approximation of some normed spaces by Riemannian manifolds is possible by using periodic Riemannian metrics with a period diameter going to zero. Consider for instance the sequence (Rn, k−1dg)k≥1 where dg is a fixed periodic Riemannian distance. It con- verges to Rn equipped with its “stable norm” as defined for instance in [9, section 8.5.2]. It is not clear whether any norm may be attained in this way and this question is related to the notorious open problem of characterizing the stable norms [10]. Finally, note that it is impossible to approximate a non-Hilbertian normed space in Gromov Hausdorff topology by Riemannian manifolds with non-negative Ricci curvature. This is because any such limit metric measure space that contains a line has to split as a product of R and another metric measure space by the splitting theorem for Ricci limit spaces established by Cheeger and Colding [12], see also [32, Conclusions and open problems]. This argument also applies to the Heisenberg group.
Moreover it is proven in [19] that (H, dcc) also cannot be approximated by a sequence of Riemannian manifolds with any uniform lower bound on the Ricci curvature.
3. Gigli–Mantegazza flow starting from a cone
In this section we will analyse the construction in the case where the initial datum is an Euclidean cone. More precisely, we will consider the cones of angle π and π/2. We will show that for all times t the resulting metric dt retains a warped product form in both cases. In the first case, it has a conic singularity of angle √
2π at the apex for all t. In the second case, the asymptotic angle at the apex is zero for all t. Thus in these natural examples, the flow does not smoothen out the singularity.
In Sections 3.1 to 3.3 we will present the case of the cone of angleπin detail.
For the cone of angle π/2 we will state the main results in Section 3.4 and omit part of the proofs, since the arguments are very similar.
3.1. Preliminaries. We will first recall basic properties of Euclidean cones and give an explicit representation of the heat kernel on the cone of angle π in the sense of RCD spaces. Moreover, we will exhibit a convenient way to calculate Wasserstein distances in the cone, via a lifting procedure from the cone to R2.
3.1.1. Euclidean cones and optimal transport. The Euclidean coneC(θ) with angle θ∈[0,2π] is defined as the quotient
C(θ) =
[0,∞)×[0, θ]
.
∼,
where we write (r, α) ∼(s, β) if and only if r =s= 0 or |α−β| ∈ {0, θ}.
The cone distance dis given by d r, α),(s, β)
= q
r2+s2−2rscos min |α−β|, θ− |α−β|
, which is well defined on the quotient. Note that the cone without the apex, i.eC(θ)\{o}, whereois the equivalence class of (0,0), is an open Riemannian manifold with the metric tensor (dr)2 +r2(dα)2. Its geometry is locally Euclidean. The associated Riemannian distance is the cone distance and the distance on the full cone C(θ) is its metric completion.
We will be concerned in particular with the cone of angleπ. In this case we have the alternative characterization as the quotient
C(π) =R2 σ ,
where the map σ :R2 → R2 is the reflection at the origin, i.e. σ(x) = −x.
Let us denote by P :R2 → C(π) the canonical projection. Then the cone distance between p, q∈C(π) can be written as
d(p, q) = min |x−y|,|x+y|
,
where x, y∈ R2 are such that P(x) = p, P(y) =q. The Hausdorff measure onC(π) is given asm= 12P#Leb, where Leb denotes the Lebesgue measure on R2.
Now, we show how to calculate efficiently Wasserstein distance in the cone C(π). We will denote by WR2 and WC(π) the L2 transport distances on R2 and C(π) built from the Euclidean distance and the cone distance d respectively. If no confusion can arise we shall simply write W.
Let us introduce the set of measures onR2 with finite second moment, that are symmetric with respect to the origin. We set
P2sym(R2) ={µ∈P2(R2) : σ#µ=µ}.
Note that given a measure ν ∈ P2(C(π)) there exists a unique measure L(ν) ∈ P2sym(R2) such that P#L(ν) = ν. We call L(ν) the symmetric lift of ν.
We have the following useful fact.
Lemma 3.1. For any two measuresµ, ν ∈P2(C(π))it holds WC(π)(µ, ν) =WR2(L(µ), L(ν)).
In other words, the mappingP2sym(R2)→P2(C(π)), µ7→P#µis an isom- etry. Moreover, for any two measures µ, ν ∈P2(R2) we have
WC(π)(P#µ, P#ν)≤WR2(µ, ν).
Proof. Let us first prove the second statement. Let µ, ν ∈P(R2) and π a transport plan between µ and ν. Define a transport plan ¯π between P#µ and P#ν by setting ¯π = (P ⊗P)#π. Therefore,
Z
|y−x|2 dπ(x, y)≥ Z
d(P(y), P(x))2 dπ(x, y) = Z
d2 d¯π . (3.1) Taking the infimum overπ, we get the second statement. We turn now to the first statement. Let µ, ν ∈P2sym(R2) and let ¯π be a transport plan between P#µ andP#ν. We can find a measurable map Q:C(π)×C(π)→R2×R2 such that (P⊗P)◦Q= Id and|x−y|=d(p, q) for Q(p, q) = (x, y). These properties also hold for−Qthat we noteQ−. The marginals of the transport plan π= 12(Q#π¯+Q−#π) are symmetric, hence they coincide with¯ µand ν.
Moreover π is concentrated on the set {(x, y) ∈ R2×R2, d(P(x), P(y)) =
|y−x|} so that we have equality in (3.1). Taking the infimum over ¯π and taking into account the second statement, we obtain the first statement.
3.1.2. RCD structure and the heat kernel. Here we verify that the coneC(π) fits into the framework of RCD spaces considered in [17] and we give an explicit description of the heat kernel in this case.
Indeed, the metric measure space (C(π), d, m) satisfies the condition RCD(0,∞) as proven for instance in [21, Thm. 1.1]. In order to identify the heat semi- group Ht acting on measures and the heat kernel Htδx in this example, it is sufficient to exhibit an explicit solution to the Evolution Variational In- equality using [4, Thm. 5.1], see Section 2.3. This will be done again via the lifting to R2.
We denote byγxt the Gaussian measure with variance 2tcentered atx∈R2: γxt(dy) = 1
4πtexp
−|y−x|2 4t
dy .
The heat semigroup in R2 acting on measures is denoted by HtR2. More precisely, for any µ∈P2(R2) we set HtR2µ(dx) =R
γyt(dx) dµ(y).
Now, putνpt=P#(γxt) wherexis such thatP(x) =p. We define a semigroup HtC(π) acting on P(C(π)) via
HtC(π)µ(dq) = Z
νpt(dq) dµ(p). Note that we have HtC(π)=P#◦HtR2 ◦L.
Lemma 3.2 (Evolution Variational Inequality). For everyµ, χ∈P2(C(π)) such that Ent(χ)<∞ and every t≥0 we have
1
2WC(π)2 (HtC(π)µ, χ)−1
2WC(π)2 (µ, χ)≤t
Ent(χ)−Ent(HtC(π)µ) . Proof. Let L(µ), L(χ) ∈P2σ(R2) be the lifts ofµ, χ. Note that HtR2L(µ) is the symmetric lift of HtC(π)µ. SinceHtR2 satisfies the Evolution Variational Inequality, see e.g. [2, Thm. 11.2.5], we find
1 2W2
R2(HtR2L(µ), L(χ))− 1 2W2
R2(L(µ), L(χ))
≤t
Ent(L(χ))−Ent(HtR2L(µ)) .
Observing that Ent(L(µ)) = Ent(µ) for anyµ∈P2(C(π)) and its symmet- ric lift L(µ) and using Lemma 3.1, this immediately yields the claim.
In view of [4, Thm. 5.1], this shows again that (C(π), d, m) satisfies RCD(0,∞) and that HtC(π) is the associated heat semigroup. In particular, νpt = HtC(π)δp is the heat kernel at timetcentered atp.
We finish this section by noting the following contraction property of the heat flow:
WC(π)(νpt, νqt)≤d(p, q), ∀p, q∈C(π), t≥0. (3.2)
Indeed, choosing x, y with P(x) = p, P(y) = q and d(p, q) = |x−y|, by Lemma 3.1 and convexity of the squared Wasserstein distance we have
WC(π)(νpt, νqt) =WR2
1
2(γxt +γ−xt ),1
2(γyt+γ−yt )
≤WR2(γxt, γty) =|x−y|=d(p, q).
3.1.3. Absolutely continuous curves and the continuity equation. We recall the characterization of absolutely continuous curves in the Wasserstein space of the Euclidean spaces via solutions to the continuity equation. Moreover, we formulate a convenient estimate on the driving vector field in the conti- nuity equation.
Proposition 3.3 ([2, Thm. 8.3.1]). A weakly continuous curve (µs)s∈[0,T]
in P2(Rn) is 2-absolutely continuous with respect to W if and only if there exists a Borel family of vector fields Vs with RT
0 kVsk2L2(µs;Rn) ds <∞ such that the continuity equation
∂sµ+ div(µsVs) = 0
holds in distribution sense. In this case we have |µ˙s| ≤ kVskL2(µs;Rn) for a.e. s. Moreover,Vs is uniquely determined for a.e. s if we require
Vs∈TµsP2(Rn) :={∇ψ | ψ∈Cc∞(Rn)}L
2(µs;Rn)
and it holds |µ˙s|=kVskL2(µs;Rn).
The next lemma states a simple condition for existence and uniqueness of solutions to the continuity equation.
Lemma 3.4. Let µ∈P2(Rn) with strictly positive Lebesgue density ρ and assume that µ satisfies the Poincar´e inequality
Z
|f|2dµ≤C Z
|∇f|2 dµ , for all f ∈ Cc∞(Rn) with R
f dµ = 0. Let s ∈ L1(Rn,Leb) be such that R s= 0 and
ks/√
ρk2L2 =
Z s2(x)
ρ(x) dx < ∞.
Then there exists a unique vector field V ∈TµP2(Rn)such that the equation s+ div(µV) = 0
holds in distribution sense. Moreover, we have kVk2L2(µ;Rn) =
Z
|V|2 dµ ≤ Cks/√
ρk2L2 . (3.3) Proof. For any f ∈Cc∞(Rn) with R
f dµ= 0 we deduce from the Cauchy–
Schwarz and Poincar´e inequalities that the bilinearB :f 7→R
sf satisfies B(f)≤
Z s2 ρ
12Z f2ρ
12
≤ ks/√ ρkL2
√ C
Z
|∇f|2 dµ 12
.
Thus, identifying f with its gradient, the map B can be extended to a bounded linear functional on the Hilbert space Tµ := TµP2(Rn) equipped with the scalar product
hU, WiL2(µ;Rn)= Z
U ·W dµ . Moreover, the norm of B is bounded by √
Cks/√
ρkL2. Thus, by the Riesz representation theorem there exists a unique vector field V ∈Tµ such that B(W) =hV, WiL2(µ;Rn) and kVkL2(µ;Rn)≤√
Cks/√
ρkL2. In particular, for any f as above we have
Z
sf = B(f) = Z
V · ∇f dµ .
Thus V is the unique distributional solution tos+ div(µV) = 0 in Tµ. 3.2. Warped structure of the convoluted cone. Having identified the heat kernel in Lemma 3.2, we can now analyse in detail the construction of [17] in the case of C(π). Let us defineιt:C(π)→P2(C(π)) via ιt(p) =νpt. This map is obviously injective and by (3.2) satisfies Assumption 2.1. Thus, as outlined in Section 2 we introduce
det(p, q) =WC(π)(νpt, νqt)
and definedtto be the associated length distance as in (2.2). Recall that the use of the the heat equation is supposed to produce a kind of convolution for metric spaces. The rotational symmetry of C(π) is preserved by this transformation so that the resulting space will retain a warped structure.
We first give a partial converse to the Lipschitz estimate (2.1).
Lemma 3.5. For any t≥0 and r >0 there exists a constant C(t, r) such that for all p, q∈C(π)\Br:
C(t, r)d(p, q)≤det(p, q), (3.4) where Br={p∈C(π) :d(o, p)≤r}.
In particular, in view of Proposition 2.4 this shows that det and dt induce the same topology as the cone distance on C(π).
Proof. Letx, y∈R2 such thatd(p, q) =|x−y|. Without restriction we can assume that |x| ≤ |y| and that x = (x1,0), y = (y1, y2) with y1, y2 ≥ 0.
Let A be the line passing through the origin at angle 3π/8 with the first coordinate axis and let prA denote the orthogonal projection ontoL. Then, setting µtx= 12(γtx+γ−xt ), we have
det(p, q) =WR2(µtx, µty)≥WR1((prA)#µtx,(prA)#µty)
Note that (prA)#µtx is the mixture of two one-dimensional Gaussians with variance 2t and centers ±prAx. Note further that |prA(x)−prA(y)| ≥ cos(3π/8)|x−y| since the angle of A with y−x is less than 3π/8. Thus it suffices to establish the following claim: For any t0 ≥0 andr > 0 there exists a constant C(t0, r) such that for all t≤t0 and x, y≥r:
C(t0, r)|x−y| ≤WR 1
2(γxt+γ−xt ),1
2(γyt+γ−yt )
, (3.5)
where by abuse of notation γxt denotes also the one-dimensional Gaussian measure with variance 2tand center x. By convexity of WR2 the right hand side is decreasing in t. Thus, by scaling it suffices to consider t = 1. In dimension 1, the optimal transport plan is known to be the monotonic re- arrangement. The two measures in (3.5) are symmetric so that the mass on R+is mapped onR+. Observe that the measureγx1+γ−x1 restricted toR+is distributed asω#γx1 whereω:x7→ |x|. Hence the right hand side of (3.5) is WR(ω#γx1, ω#γy1). Applying Jensen’s inequality in the definition of WR we see that the distance between the means of these measures is a lower bound.
But the mean of ω#γx1 is R
|s|dγx1(s) =R
|s−x|dγ10(s). As a function of x∈R+, this is a strictly convex function with derivative zero at zero on the right and tangent to the first bisector at +∞. In fact the second derivative in distribution sense is 2γ01. The estimate (3.5) follows from these remarks
together, provided x, y≥r for somer >0.
Next, we show that in the definition of dt we can restrict the infimum to Lipschitz curves with respect to the cone distance d.
Lemma 3.6. For any t≥0 andp, q∈C(π) we have dt(p, q) = inf
Z T
0
|p˙s|tds
, (3.6)
where the infimum is taken over all d-Lipschitz curves (ps)s∈[0,T] such that p0 =p, pT =q and |p˙s|t denotes the metric derivative with respect to det. If p6=oand q 6=o, one can restrict to the curves supported in C(π)\ {o}.
Thus the construction of dt given here is consistent with the general con- struction in RCD spaces given in [17] (see Remark 2.2).
Proof. The inequality “≤” follows immediately from the fact that any d- Lipschitz curve is also det-Lipschitz by (3.2). To see the reverse inequality, first recall that by Remark 2.3 we can restrict the infimum in (2.2) to det- Lipschitz curves. Then the statement follows from Lemma 3.5. Indeed, given ε >0, let (ps)s∈[0,T] be a det-Lipschitz curve such that
Z T
0
|p˙s|tds≤dt(p, q) +ε .
Recall that (ps) is d-continuous. If it avoids the origin (and thus also a neighborhood around it) then (ps) is alsod-Lipschitz by (3.4). If the curve hits the origin, put for sufficiently small r >0:
s1:= inf{s∈[0, T] :ps∈Br}, s2 := sup{s∈[0, T] :ps∈Br}. We can construct a d-Lipschitz curve (pes) by replacing the part (ps)s∈[s1,s2] with a piece of circle connecting ps1 to ps2. From (3.2) we see that the det- length of (pes) is bounded by dt(p, q) +ε+πr. Choosingr sufficiently small and using the arbitrariness of εwe obtain the inequality “≥” in (3.6).
In the case p = o or q = o, the ray from or to the apex is a minimizing
curve. Note that it is a d-Lipschitz curve.
Let us further observe the particular behavior of the distance under scaling of space and time.
Lemma 3.7. For any t >0 andp, q∈C(π) we have dt(p, q) =√
t·d1 √
t−1p,√ t−1q
. (3.7)
Here, for λ≥0 and p= (r, α)∈C(π) we set λp= (λr, α).
Proof. It suffices to establish the identity (3.7) with dt replaced by det. It then passes easily to the associated length distance. Recall that det(p, q) = W2,R2(12(γtx+γ−xt ),12(γty+γ−yt ) forx, ysuch thatP(x) =p, P(y) =q. Intro- duce the dilation sλ :x 7→ λx and note that γtx = (s√t)#γ√1
t−1x. Now the
claim is immediate.
We have the following result on the metric structure of the convoluted cone.
Proposition 3.8. The distance dt is induced by a metric tensor gt on the open manifold C(π)\ {o} which is of warped product form
gt(r,α)(·,·) =R(r/√
t)dr2+r2A(r/√
t)dα2, (3.8)
where R, A: (0,∞) → (0,1] are bounded functions. Moreover, the distance dt on the full cone is obtained for p0, p1∈C(π) by
dt p0, p1
= inf Z 1
0
q
R(rs/√
t)|r˙s|2+r2sA(rs/√
t)|α˙s|2 ds , (3.9) where the infimum is taken over all Lipschitz curves (ps)s∈[0,1] of (C(π), d) connecting p0, p1 and |r˙s|,|α˙s| denote the metric derivatives of the polar coordinates of ps.
In (3.8) R and A stand for radial and angular.
Proof. Recall from Section 2 that dt(p, q) = inf
(ps)
Z T
0
|ν˙pts|ds ,
where νpt = ιt(p) and |ν˙pts| denotes the metric derivative with respect to WC(π). We will first use the lifting to R2 and the characterization of the Wasserstein metric derivative in terms of solutions to the continuity equation to relatedtto a smooth metric tensor onR2and then we will push this tensor to the cone to obtain the desired warped structure.
From Lemma 3.7 we immediately infer that it is sufficient to considert= 1.
For brevity let us set µx= 12γx+12γ−x and letfx be its density, i.e.fx(y) =
1
2η(y−x) + 12η(y+x), where η(y) = 4π1 exp(−|y|2/4) is the density of the 2-dimensional Gaussian at time 1.
We define a metric tensor eg onR2 by setting forx, w ∈R2: egx(v, w) =
Z
hVxv, Vxwidµx , where Vxw is the unique vector field inTµxP(R2) solving
d dh
h=0fx+hw = 1
2w· ∇η(x+y)− ∇η(x−y)
= −div(µxVxw) (3.10) given by Lemma 3.4 (applied to s = dhdfx+hw and µ = µx). Indeed, by uniqueness,Vxw depends linearly on w, hence gx(v, w) is a bilinear form.
Now, define a metric tensor g on the open manifold C(π)\ {o} by setting for p= (r, α)∈C(π)\ {o} and v, θ∈R:
gp (v, θ),(v, θ)
=egx(w, w), where
x=r cosα
sinα
, w=v cosα
sinα
+θr
−sinα cosα
,
and extend via polarization. That g takes the form (3.8) is a consequence of the fact that egx(w, w) is invariant under reflecting w at the line passing through the origin andx, implying thatgp (v, θ),(v, θ)
=gp (v,−θ),(v,−θ) , and the invariance of egx(w, w) under simultaneous rotation ofx, w. Explic- itly, we have
R(r) =egr(1,0) (1,0),(1,0)
, A(r) =egr(1,0) (0,1),(0,1)
. (3.11) Let us now prove (3.9), i.e. that d1 is induced by the tensorg. By Lemma 3.6 we have
d1(p0, p1) = inf Z 1
0
|p˙s|1 ds ,
where|p˙s|1 is the metric derivative ofpswith respect to the distance de1 and the infimum is over Lipschitz curves in (C(π), d). Let us consider a Lipschitz curve (ps)s∈[0,1] in (C(π), d) with polar coordinates (rs)s and (αs)s. Let (xs)s∈[0,1]be a continuous a curve such thatP(xs) =ps. By (3.2) the curves νp1s and µxs are Lipschitz with respect to WC(π) and WR2 respectively and by definition of de1 and Lemma 3.1 we have
|p˙s|1 =|ν˙p1s|=|µ˙xs|,
where the latter two metric derivatives are calculated with respect to WC(π) and WR2 respectively. By the characterization of absolutely continuous curves, there exists for a.e. sa vector field Vs ∈ TµxsP(R2) such that the continuity equation ∂sµxs = −div(µxsVs) holds in distribution sense. But for a.e. sthe left hand side is given by 12ws· ∇η(·+xs)− ∇η(· −xs)
with ws = ˙rs
cosαs
sinαs
+ ˙αsrs
−sinαs
cosαs
.
Hence, the uniqueness statement in Lemma 3.4 implies that for a.e. s we have Vs=Vxwss and thus
|p˙s|21 =egxs(ws, ws) =gps ( ˙rs,α˙s),( ˙rs,α˙s)
=|r˙s|2R(rs) +r2s|α˙s|2A(rs). This yields that dt(p0, p1) is given by the right hand side in (3.9).
Finally, we turn to the boundedness of R and A. In fact for w a vector of R2 the vector field V : y 7→ λx(y)w+ (1−λx(y))(−w) where λx(y) = [η(y−x)/(η(y−x) +η(y+x))] satisfies (3.10) in place ofVxw. It is an ele- ment of L2(µx;Rn) with norm smaller than or equal to|w|. The orthogonal projection on TµxP(R2) contracts the norm and provides another solution to (3.10). According to the uniqueness statement in Lemma 3.4 it is Vxw. Hence we have proved egx(w, w) ≤ |w|2. It follows that the functions A and R defined in (3.11) are bounded from above by 1.
Remark 3.9. We believe that the functions R and A in the proposition above are smooth, so that dt would be induced by a smooth metric tensor on C(π)\ {o}. From the explicit expression (3.16) for R given below in the proof of Theorem 3.11, it is readily checkedRis smooth. Proving smoothness for Aseems non-trivial due to the non-compactness of the cone.
Proposition 3.10. As t goes to zero, the metric space (C(π), dt) tends to (C(π), d) pointwise and in the pointed Gromov–Hausdorff topology.
Proof. By construction of dt and by the contractivity (3.2) with (2.3) we have the chain of inequalities
det≤dt≤d .
From Proposition 2.4 we already know that det pointwise converges to d as t → 0 whence the convergence of dt follows. The convergence in pointed Gromov–Hausdorff topology follows as in the proof of Proposition 2.4.
As an immediate consequence of Proposition 3.8, we deduce that a dt- minimizing curve connecting the apex o= (0,0) to the point (r,0)∈C(π) is given by the curve (sr,0)s∈[0,1]. Hence the distance of (r,0) from the apex o is
dt (r,0), o
= Z 1
0
r q
R sr/√ t
ds= Z r
0
r R
s/√ t
ds . (3.12) 3.3. Persistence of the conic singularity for C(π). We will show that the new distance dt has a conic singularity at the origin of angle√
2π inde- pendent of t. In order to do so we will compare for smallr the distance of a point pr = (r,0) from the origin to the length of a circle around the origin passing through pr.
More precisely, for r >0 setρt(r) =dt (0, r), o
and define lt(r) =
Z π
0
|p˙rs|tds .
where the curve pr: [0, π]→C(π) is given by prs = (r, s).
Theorem 3.11. For each t >0 we have
r→0lim lt(r) ρt(r) =√
2π . In other words, the angle at the apex o is√
2π. In particular, a singularity persists at o. With the notation of Proposition 3.8 we have more precisely R(r)∼r2/2 andA(r)∼r2/4 as r→0.
Moreover, C(√
2π)is both, the tangent space of(C(π), dt)ato, and the limit in the pointed Gromov–Hausdorff topology as t goes to infinity.
Remark 3.12. The discontinuity at t= 0 of the asymptotic angle atomight seem intriguing at first in view of the convergence of dt to the original distance d given by Proposition 3.10. Note however, that the asymptotic angle is in a certain sense a first order quantity, while the convergence of distances is zero order. Intuitively, the discontinuity can be understood from the scaling property (3.7). After zooming in at scale r, the heat kernel
measure at a very small time t looks like the heat kernel measure at the larger time t/√
r at the original scale.
Proof. We will calculateρt andlt asymptotically asr →0.
From Proposition 3.8 and (3.12) we have ρt(r) =
Z 1 0
r q
R sr/√ t
ds (3.13)
lt(r) =πr q
A r/√ t
. (3.14)
Thus, it remains to calculate R and A. Denote by fx the density of µ1x =
1
2γx1+12γ−x1 . We setxr = (r,0)∈R2and recall from the proof of Proposition 3.8 that
R(r) =||Vx(1,0)r ||2L2(µ1xr;R2), A(r) =||Vx(0,1)
r ||2L2(µ1xr;R2), where the vector field Vxw ∈Tµ1
xP(R2) is defined uniquely by the continuity equation
d dh
h=0fx+hw = −div(fxVxw). (3.15) Note that
fx(y) = 1/2 η(y1−x1)η(x2−y2) +η(x1+y1)η(x2+y2) ,
where η denotes y 7→ (4πt)−1/2exp(−y2/4t), the 1-dimensional Gaussian density at time 1. Let us first concentrate on R. Here, we have to solve
d dh
h=0fxr+h(1,0)(y) =η(y2)1
2 −η0(y1−r) +η0(y1+r)
=−div
fxrVx(1,0)r (y). It is easily checked that the solution is given by
Vx(1,0)r (y) = 1
0
η(y1−r)−η(y1+r) η(y1−r) +η(y1+r) . which indeed belongs to Tµ1
xrP(R2). Thus, we have R(r) =
Z
|Vx(1,0)r |2 dµ1xr = 1 2
Z Z
η(y2)|η(y1−r)−η(y1+r)|2
η(y1−r) +η(y1+r) dy1 dy2
= 1 2
Z |η(y1−r)−η(y1+r)|2
η(y1−r) +η(y1+r) dy1. (3.16)
To determine the asymptotic behavior as r →0, we first note that R(r) = 1
2r2
Z |2η0(y1)|2
2η(y1) dx+o(r2) =r2 Z y21
4 η(y1) dy1+o(r2) = r2
2 +o(r2). (3.17)
